2005/02/25 by Igor Rivin, Rivin, Igor
Mathematics · #52B10 #52B11 #57M50 #FOS: Mathematics #Geometric Topology (math.GT) #History and Theory of Mathematics #Mathematics and Applications #Metric Geometry (math.MG) #Point processes and geometric inequalities #math.GT #math.MG #msc:52B10 #msc:52B11 #msc:57M50
paper · pdf · doi:10.48550/arxiv.math/0502543
12 pages; revision has minor cosmetic changes
openalex publication_date 2005/02/25 · arxiv created 2005/03/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of allowable angles. A proof of this has recently been given by F. Luo (see math.GT/0412208). In this paper we give a simple proof of this conjecture, prove much sharper regularity results, and then extend the method to apply to a large class of convex polytopes. The simplex argument works without change in dimensions greater than 3 (and for spherical simplices in all dimensions), so the bulk of this paper is concerned with the three-dimensional argument. The estimates relating the diameter of a polyhedron to the length of the systole of the polar polyhedron are of independent interest.